A heating coil with a surface area of $150$ cm$2$

$,$ is used to heat up a $10$ L water bath during a

lab experiment. The whole surface area of the heating coil is immersed in the water bath.

The density of water is $1$ kg L$-1$ and is considered to be constant and independent of

temperature. The temperature of the heating coil, Thc$,$ is $95\backslash$deg C and it remains constant,

whereas the temperature of the water at the beginning of the experiment, Two, is $20\backslash$deg C and it

rises to T$.$ The overall heat transfer coefficient is $4$ J min$-1$ cm$-2$ K$-1$ and it is independent of

temperature. The heat capacity of water is $4185\mathrm{.}5$ J kg$-1$ K$-1$

$.$

$($a$)$ Write an energy balance on the water in the bath, in the form of dT$/$dt and provide an

initial condition. $[9$ marks$]$

$($b$)$ Develop an expression for T$($t$)$ and check if the solution satisfies the initial condition.

$[11$ marks$]$

$($c$)$ How long will it take for the water to heat up to $50\backslash$deg C$?$